Suppose a firecracker explodes at time t. The experimenters in reference frame S determine that the explosion happened at position x. Similarly, the experimenters in $\text{S}\prime$ find that the firecracker exploded at $x\prime$ in their reference frame. What is the relationship between x and $x\prime ?$

FIGURE 36.2 shows the explosion and the two reference frames. You can see from the figure that $x=x\prime +vt$, thus

FIGURE 36.2 The position of an exploding firecracker is measured in reference frames S and S$\prime$.

These are the Galilean transformations of position. If you know a position measured by the experimenters in one inertial reference frame, you can calculate the position that would be measured by experimenters in any other inertial reference frame.

Suppose the experimenters in both reference frames now track the motion of the object in FIGURE 36.3 by measuring its position at many instants of time. The experimenters in S find that the object’s velocity is $\vec{u}$. During the same time interval $\text{Δ}t$, the experimenters in $\text{S}\prime$ measure the velocity to be $\vec{u}\prime$.

FIGURE 36.3 The velocity of a moving object is measured in reference frames S and S$\prime$.

We can find the relationship between $\vec{u}$ and $\vec{u}\prime$ by taking the time derivatives of Equations 36.1 and using the definition $u_x=dx/dt:$

The equation for $u_z$ is similar. The net result is

Equations 36.2 are the Galilean transformations of velocity. If you know the velocity of a particle in one inertial reference frame, you can find the velocity that would be measured by experimenters in any other inertial reference frame.

EXAMPLE 36.1 The speed of sound

An airplane is flying at speed $200\text{m}/\text{s}$ with respect to the ground. Sound wave 1 is approaching the plane from the front, sound wave 2 is catching up from behind. Both waves travel at $340\text{m}/\text{s}$ relative to the ground. What is the speed of each wave relative to the plane?

MODEL Assume that the earth (frame S) and the airplane (frame $\text{S}\prime$) are inertial reference frames. Frame $\text{S}\prime$, in which the airplane is at rest, moves at $v=200\text{m}/\text{s}$ relative to frame S.

VISUALIZE FIGURE 36.4 shows the airplane and the sound waves.

FIGURE 36.4 Experimenters in the plane measure different speeds for the waves than do experimenters on the ground.

FIGURE 36.4 Full Alternative Text

SOLVE The speed of a mechanical wave, such as a sound wave or a wave on a string, is its speed relative to its medium. Thus the speed of sound is the speed of a sound wave through a reference frame in which the air is at rest. This is reference frame S, where wave 1 travels with velocity $u_1=-340\text{m}/\text{s}$ and wave 2 travels with velocity $u_2=+340\text{m}/\text{s}$. Notice that the Galilean transformations use velocities, with appropriate signs, not just speeds.

The airplane travels to the right with reference frame $\text{S}\prime$ at velocity v. We can use the Galilean transformations of velocity to find the velocities of the two sound waves in frame $\text{S}\prime :$

$$\begin{array}{l}{u}_1^{\prime }=u_1-v=-340\text{ m/s}-200\text{ m/s}=-540\text{ m}/\text{s }\\ u_2^{\prime }=u_2-v=340\text{ m/s}-200\text{ m/s}=140\text{ m}/\text{s}\end{array}$$

REVIEW This isn’t surprising. If you’re driving at 50 mph, a car coming the other way at 55 mph is approaching you at 105 mph. A car coming up behind you at 55 mph is gaining on you at the rate of only 5 mph. Wave speeds behave the same. Notice that a mechanical wave appears to be stationary to a person moving at the wave speed. To a surfer, the crest of the ocean wave remains at rest under his or her feet.